Vito Prosciutto: Teaching community college math on the road to a PhD.

Monday, October 24, 2005

A question for algebraic geometry types 

Is there a higher-dimensional analog for Bezout's theorem? (To clarify, since there seem to be at least three theorems that go by this name, what I'm refering to is this:
Given two curves f(x,y)=0 and g(x,y)=0 in CP2, if there are no common components, the number of points of intersection of the two curves (including multiplicity) is deg(f)deg(g)
My intuitive sense is that at higher dimensions, instead of counting points, we count the degrees of the curves generated by the intersections. e.g., in three dimensions, the intersection of two planes will be a line (1*1=1), the intersection of a plane and a 2nd degree surface will be a 2nd degree curve or two lines. I suppose tangent points probably turn into nice surfaces once we look beyond R3 to C3. But it seems sufficiently basic that surely someone has already mapped this out and proven it.

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